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Mathematical and Computer Modelling of Dynamical Systems
Methods, Tools and Applications in Engineering and Related Sciences
Volume 30, 2024 - Issue 1
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Research Article

Model Reduction of Parametric Differential-Algebraic Systems by Balanced Truncation

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Pages 303-341 | Received 10 Sep 2021, Accepted 05 Mar 2024, Published online: 11 Jul 2024

References

  • Alì G, Banagaaya N, Schilders WHA, Tischendorf C. 2014. Index-aware model order reduction for differential-algebraic equations. Math Comput Model Dyn Syst. 20(4):345–373. doi:10.1080/13873954.2013.829501.
  • Antoulas AC. 2005. Approximation of Large-Scale Dynamical Systems. In: Advance Des Control Vol. 6 Philadelphia, PA: SIAM Publications, doi:10.1137/1.9780898718713.
  • Antoulas AC, Gosea IV, Heinkenschloss M. 2020. Data-driven model reduction for a class of semi-explicit DAEs using the Loewner framework. In: Reis T and Ilchmann A, editors. Progress in Differential-Algebraic Equations II, Differ.-Algebr. Equ. Forum. Cham: Springer; p. 185–210. doi: 10.1007/978-3-030-53905-4_7.
  • Banagaaya N. 2014. Index-Aware Model Order Reduction Methods for DAEs. The Netherlands: Proefschrift, Technische Universiteit Eindhoven. doi:10.6100/IR780937.
  • Banagaaya N, Ali G, Schilders WHA. 2016. Index-aware Model Order Reduction Methods. In: Atlantis Studies Science Computer Electromagnet. Paris: Atlantis Press; Vol. 2.
  • Bartels RH, Stewart GW. 1972. Solution of the matrix equation AX+XB=C: Algorithm 432. Comm ACM. 15(9):820–826. doi:10.1145/361573.361582.
  • Baur U, Beattie CA, Benner P, Gugercin S. 2011. Interpolatory projection methods for parameterized model reduction. SIAM J Sci Comput. 33(5):2489–2518. doi:10.1137/090776925.
  • Baur U, Benner P, Feng L. 2014. Model order reduction for linear and nonlinear systems: A system-theoretic perspective. Arch Comput Methods Eng. 21(4):331–358. doi:10.1007/s11831-014-9111-2.
  • Beattie CA, Gugercin S, Mehrmann V. 2022. Structure-preserving interpolatory model reduction for port-Hamiltonian differential-algebraic systems. In: Beattie C, Benner P, Embree M, Gugercin S, and Lefteriu S, editors. Realization and Model Reduction of Dynamical Systems. Cham: Springer; p. 235–254. doi:10.1007/978-3-030-95157-3_13.
  • Benner P, Gugercin S, Willcox K. 2015. A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4):483–531. doi:10.1137/130932715.
  • Benner P, Kürschner P, Tomljanović Z, Truhar N. 2016. Semi-active damping optimization of vibrational systems using the parametric dominant pole algorithm. ZAMM Z Angew Math Mech. 96(5):604–619. doi:10.1002/zamm.201400158.
  • Benner P, Ohlberger M, Cohen A, Willcox K. 2017. editors, Model Reduction and Approximation: Theory and Algorithms. Comput. Sci. Eng. SIAM. doi:10.1137/1.9781611974829.
  • Benner P, Saak J, Uddin MM. 2016. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numer Algebra Cont Optim. 6(1):1–20. doi:10.3934/naco.2016.6.1.
  • Benner P, Schneider A. 2013. Balanced truncation for descriptor systems with many terminals. Max Planck Institute Magdeburg Preprints MPIMD/13–17, Available at https://csc.mpi-magdeburg.mpg.de/preprints/2013/MPIMD13-17.pdf.
  • Benner P, Stykel T. 2017. Model order reduction for differential-algebraic equations: A survey. In: Ilchmann A and Reis T, editors. Surveys in Differential-Algebraic Equations IV, Differ.-Algebr. Equ. Forum. Cham: Springer; p. 107–160. doi:10.1007/978-3-319-46618-7_3.
  • Benner P, Tomljanović Z, Truhar N. 2011. Dimension reduction for damping optimization in linear vibrating systems. ZAMM Z Angew Math Mech. 91(3):179–191. doi:10.1002/zamm.201000077.
  • Berger T, Ilchmann A, Trenn S. 2012. The quasi-Weierstraß form for regular matrix pencils. Linear Algebra Appl. 436(10):4052–4069. doi:10.1016/j.laa.2009.12.036.
  • Brenan KE, Campbell SL, Petzold LR. 1995. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Philadelphia, PA: SIAM. doi:10.1137/1.9781611971224.
  • Campbell SL. 1980. Singular Systems of Differential Equations. In: Research Notes in Mathematics Vol. 40. San Francisco: Pitman Advanced Publishing Program, doi:10.1002/nme.1620150916.
  • Campbell SL, Meyer CD, Rose NJ. 1976. Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. Comput Math Appl. 31(3):411–425. doi:10.1137/0131035.
  • Druskin V, Simoncini V. 2011. Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst Cont Lett. 60(8):546–560. doi:10.1016/j.sysconle.2011.04.013.
  • Eid R, Castañé-Selga R, Panzer H, Wolf T, Lohmann B. 2011. Stability-preserving parametric model reduction by matrix interpolation. Math Comput Model Dyn Syst. 17(4):319–335. doi:10.1080/13873954.2011.547671.
  • Flagg G, Beattie C, Gugercin S. 2012. Convergence of the iterative rational Krylov algorithm. Syst Cont Lett. 61(6):688–691. doi:10.1016/j.sysconle.2012.03.005.
  • Geuss M, Panzer H, Lohmann B. On parametric model order reduction by matrix interpolation. In Proceedings of the 12th European Control Conference, 3433–3438, Strasbourg, France, 2013. doi:10.23919/ECC.2013.6669829.
  • Glover K. 1984. All optimal Hankel-norm approximations of linear multivariable systems and their L∞ -error bounds. Internat J Control. 39(6):1115–1193. doi:10.1080/00207178408933239.
  • Gosea IV, Poussot-Vassal C, Antoulas AC. 2021. Data-driven modeling and control of large-scale dynamical systems in the Loewner framework: methodology and applications. In: Zuazua E and Trelat E, editors. Numerical Control: Part A, Hadb Numer Anal. Elsevier: North-Holland.
  • Gosea IV, Zhang Q, Antoulas AC. 2020. Preserving the DAE structure in the Loewner model reduction and identification framework. Adv Comput Math. 46(3). doi:10.1007/s10444-020-09752-8.
  • Gugercin S, Antoulas AC, Beattie C. 2008. model reduction for large-scale linear dynamical systems. SIAM J Matrix Anal Appl. 30(2):609–638. doi:10.1137/060666123.
  • Gugercin S, Stykel T, Wyatt S. 2013. Model reduction of descriptor systems by interpolatory projection methods. SIAM J Sci Comput. 35(5):B1010–B1033. doi:10.1137/130906635.
  • Hammarling SJ. 1982. Numerical solution of the stable, non-negative definite Lyapunov equation. IMA J Numer Anal. 2(3):303–323. doi:10.1093/imanum/2.3.303.
  • Heinkenschloss M, Sorensen DC, Sun K. 2008. Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations. SIAM J Sci Comput. 30(2):1038–1063. doi:10.1137/070681910.
  • Hesthaven JS, Rozza G, Stamm B. 2016. Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Cham: SpringerBriefs Math. Springer. doi:10.1007/978-3-319-22470-1.
  • Horn RA, Johnson CR. 1991. Topics in Matrix Analysis. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511840371.
  • Kunkel P, Mehrmann V. 2006. Differential-Algebraic Equations: Analysis and Numerical Solution. In: Textbooks in Mathematics. Zürich, Switzerland: EMS Publishing House.
  • Kürschner P. 2016, Efficient Low-Rank Solution of Large-Scale Matrix Equations. Dissertation, Otto-von-Guericke-Universität, Magdeburg, Germany: http://hdl.handle.net/11858/00-001M-0000-0029-CE18-2.
  • Lancaster P, Tismenetsky M. 1985. The Theory of Matrices. 2nd edition ed. Orlando: Academic Press.
  • März R. 1996. Canonical projectors for linear differential algebraic equations. Comput Math Appl. 31(4/5):121–135. doi:10.1016/0898-1221(95)00224-3.
  • Massoudi A, Opmeer MR, Reis T. 2017. The ADI method for bounded real and positive real Lur’e equations. Numer Math. 135(2):431–458. doi:10.1007/s00211-016-0805-2.
  • Mayo AJ, Antoulas AC. 2007. A framework for the solution of the generalized realization problem. Linear Algebra Appl. 425(2–3):634–662. doi:10.1016/j.laa.2007.03.008.
  • Mehrmann V, Stykel T. 2005. Balanced truncation model reduction for large-scale systems in descriptor form. In: Benner P, Mehrmann V, and Sorensen DC, editors. Dimension Reduction of Large-Scale Systems, volume 45 Lect Notes Comput Sci Eng. Berlin/Heidelberg: Springer; p. 83–115. doi:10.1007/3-540-27909-1_3.
  • Moore BC. 1981. Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans Automat Control. AC-26(1):17–32. doi:10.1109/TAC.1981.1102568.
  • Panzer H, Wolf T, Lohmann B. 2012. A strictly dissipative state space representation of second order systems at-Automatisierungstechnik. 60(7):392–397. doi:10.1524/auto.2012.1015.
  • Penzl T. 2000. A cyclic low rank Smith method for large sparse Lyapunov equations. SIAM J Sci Comput. 21(4):1401–1418. doi:10.1137/S1064827598347666.
  • Pulch R, Narayan A, Stykel T. 2021. Sensitivity analysis of random linear differential–algebraic equations using system norms. J Comput Appl Math. 397:113666. doi:10.1016/j.cam.2021.113666.
  • Quarteroni A, Manzoni A, Negri F. 2016. Reduced Basis Methods for Partial Differential Equations. In: of La Matematica per il 3+2. Cham: Springer International Publishing; Vol. 92.
  • Saak J, Köhler M, Benner P. M-M.E.S.S.-2.2 – The Matrix Equations Sparse Solvers library, February 2022. doi:10.5281/zenodo.5938237.
  • Saak J, Voigt M. 2018. Model reduction of constrained mechanical systems in M-M.E.S.S. IFAC-PapersOnLine. 51(2):661–666. doi:10.1016/j.ifacol.2018.03.112. 9th Vienna International Conference on Mathematical Modelling, Vienna, 2018.
  • Schmidt A, Wittwar D, Haasdonk B. 2020. Rigorous and effective a-posteriori error bounds for nonlinear problems – Application to RB methods. Adv Comput Math. 46:32. doi:10.1007/s10444-020-09741-x.
  • Schwerdtner P, Moser T, Mehrmann V, Voigt M. 2023. Optimization-based model order reduction of port-Hamiltonian descriptor systems. Syst Control Lett. 182:105655. doi:10.1016/j.sysconle.2023.105655.
  • Simoncini V, Druskin V. 2009. Convergence analysis of projection methods for the numerical solution of large Lyapunov equations. SIAM J Num Anal. 47(2):828–843. doi:10.1137/070699378.
  • Son NT, Gousenbourger P-Y, Massart E, Stykel T. 2021. Balanced truncation for parametric linear systems using interpolation of Gramians: a comparison of algebraic and geometric approaches. In: Benner P, Breiten T, Faẞbender H, Hinze M, Stykel T, and Zimmermann R, editors. Model Reduction of Complex Dynamical Systems, volume 171 International Series of Numerical Mathematics. Cham: Birkhäuser; p. 31–51. doi:10.1007/978-3-030-72983-7_2.
  • Son NT, Stykel T. 2017. Solving parameter-dependent Lyapunov equations using the reduced basis method with application to parametric model order reduction. SIAM J Matrix Anal Appl. 38(2):478–504. doi:10.1137/15M1027097.
  • Sorenson DC, Zhou Y. 2002. Bounds on eigenvalue decay rates and sensitivity of solutions to Lyapunov equations. CAAM Technical Reports TR02-07, Rice University, Available at https://repository.rice.edu/items/7d1381b1-adc7-4282-8486-430784cef15c
  • Stykel T. 2004. Gramian-based model reduction for descriptor systems. Math Cont Sign Syst. 16(4):297–319. doi:10.1007/s00498-004-0141-4.
  • Stykel T. 2006. Balanced truncation model reduction for semidiscretized Stokes equation. Linear Algebra Appl. 415(2–3):262–289. doi:10.1016/j.laa.2004.01.015.
  • Stykel T. 2008. Low-rank iterative methods for projected generalized Lyapunov equations. Electron Trans Numer Anal. 30:187–202. URL http://etna.mcs.kent.edu/vol.30.2008/pp187-202.dir/pp187-202.pdf
  • Tombs MS, Postlethwaite I. 1987. Truncated balanced realization of a stable non-minimal state-space system. Internat J Control. 46(4):1319–1330. doi:10.1080/00207178708933971.
  • Tomljanović Z, Beattie C, Gugercin S. 2018. Damping optimization of parameter dependent mechanical systems by rational interpolation. Adv Comp Math. 44(6):1797–1820. doi:10.1007/s10444-018-9605-9.
  • Truhar N, Tomljanović Z, Puvača M. 2019. Approximation of damped quadratic eigenvalue problem by dimension reduction. Appl Math Comput. 347:40–53. doi:10.1016/j.amc.2018.10.047.
  • Truhar N, Veselić K. 2009. An efficient method for estimating the optimal dampers’ viscosity for linear vibrating systems using Lyapunov equation. SIAM J Matrix Anal Appl. 31(1):18–39. doi:10.1137/070683052.
  • Ugrica M. Approximation of Quadratic Eigenvalue Problem and Application to Damping Optimization. Doctoral thesis, University of Zagreb, Faculty of Science, Zagreb, Croatia, 2020. URL https://urn.nsk.hr/urn:nbn:hr:217:106554.
  • Wang B, Zhang F. 1992. Some inequalities for the eigenvalues of the product of positive semidefinite Hermitian matrices. Linear Algebra Appl. 160:113–118. doi:10.1016/0024-3795(92)90442-D.
  • Zhou K, Doyle JC, Glover K. 1996. Robust and Optimal Control. Upper Saddle River, NJ: Prentice-Hall.